Problem 71 - Prealgebra

Question

Find the mistakes. Correct the right-hand side of each one. $$\frac{1}{2} \cdot \frac{3}{5}=\frac{4}{10}$$

Step-by-Step Solution

Answer

The correct right-hand side is \( \frac{3}{10} \).

Step 1: Identify the multiplication in fractions

To multiply fractions, multiply the numerators together and the denominators together. Here, multiply the numerators 1 and 3 and denominators 2 and 5.

Step 2: Multiply the numerators and denominators

The numerators are 1 and 3, and the product is 1 \( \times \) 3 = 3. The denominators are 2 and 5, so 2 \( \times \) 5 = 10.

Step 3: Write the resulting fraction

After carrying out the multiplication, the resulting fraction is \( \frac{3}{10} \).

Step 4: Compare with given right-hand side

The right-hand side of the equation should be \( \frac{3}{10} \) instead of \( \frac{4}{10} \). There was a mistake in the original calculation.

Key Concepts

Understanding Numerators and Denominators in Fractions

In any fraction, you have a numerator and a denominator separated by a division line. The numerator is the number above the line, representing the parts of the whole you have. Meanwhile, the denominator is below the line, indicating the total number of equal parts the whole is divided into.
For example, in the fraction \( \frac{3}{5} \), \( 3 \) is the numerator and \( 5 \) is the denominator. When you multiply fractions, you're essentially multiplying the numerators with each other and the denominators with each other. This fundamental understanding helps in accurately carrying out fraction operations.
It's crucial to remember that while multiplying, there is no need to match the denominators as required in addition or subtraction of fractions.

Simplifying Fractions: Why It Matters

Fraction simplification is a key step in fraction multiplication and mathematics in general. It involves making a fraction as simple as possible while keeping its value unchanged. This is done by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.

For instance, consider if the resulting fraction of a multiplication were \( \frac{8}{12} \). To simplify, find the GCD of 8 and 12, which is 4, and divide both by 4, resulting in \( \frac{2}{3} \). Simplifying fractions helps make further calculations easier and answers more intuitive. It's also useful to present your final answer in the simplest form, as it is cleaner and often more useful in real-world applications.

Correcting Mathematical Errors in Multiplication of Fractions

Mathematical errors like the one found in our original exercise are common, and catching these mistakes is vital for accuracy.
In the problem \( \frac{1}{2} \cdot \frac{3}{5} = \frac{4}{10} \), the error was in the calculation of the product of the numerators. The correct operation should yield \( \frac{3}{10} \).

• First, double-check your multiplication: \( 1 \cdot 3 eq 4 \); it should be \( 3 \).
• Next, verify the multiplication of denominators: \( 2 \times 5 = 10 \) is correct, which confirms that the error was in the numerator.

Identifying and correcting these mistakes ensures that you understand the process and helps avoid similar mistakes in future problems. Regular practice and reviewing each step can improve accuracy and confidence in handling fractions.